71
Biotechnol. Prog. 1995, 11, 71-79
A Kinetic Model for Substrate and Energy Consumption of Microbial Growth under Substrate-SufficientConditions A-P.Zeng"and W.-D. Deckwer Biochemical Engineering Division, GBF-National Research Institute for Biotechnology, Mascheroder Weg 1, D-38124 Braunschweig, Germany
The growth of heterotrophic microorganisms can be classified into substrate-limited and substrate-sufficient growth according to the relative availability of the substrate (carbon and energy source) and other nutrients. It is generally observed that the consumption rates of substrate and energy (ATP) are higher under substrate-sufficient conditions than under conditions of substrate limitation. The excess substrate and ATP consumption is often influenced by the residual concentration of substrate in a. relatively wide range. To account for these effects, a kinetic model is proposed to describe substrate and ATP consumption rates of microbial growth under substratesufficient conditions. According to the model, the specific substrate consumption rate of a substrate-sufficient culture can be expressed as the sum of the substrate consumption rate under substrate-limited conditions a t the corresponding specific growth rate and an additional consumption rate due to excess substrate. The same kinetic form also applies to the specific ATP consumption rate and t o the specific oxygen consumption rate of a n aerobic culture, respectively. The linear equations for substrate and ATP consumption rates of Pirt and of Stouthamer and Bettenhausen can be used for substrate-limited growth. The excess of substrate and ATP consumption rates at carbon surplus can be described in a form similar to that of Michaelis-Menten kinetics. The proposed kinetic model has been verified with experimental data from three continuous cultures representing both anaerobic and aerobic microbial growth on substrates with low and high degrees of reductance. Using this model, the parameters maximum growth yield and maintenance requirement (both in terms of substrate and ATP) of a culture under different growth limitations can be better defined and quantified. The range of residual substrate concentrations in which the specific rates of substrate and ATP consumption are affected can also be assessed. This information should be helpful in designing medium and reactor operating conditions and in interpreting experimental results obtained under different limiting conditions.
Introduction The optimization and control of bioprocesses often requires the establishment of a mathematical model that describes the metabolic acitivities of microorganisms, especially with respect to the responses of cells to a change in the physiological environment. Rate equations for microbial growth, substrate uptake, and product formation that describe the kinetics of a process are the basis for mathematical modeling. The rate equations used for microbial growth can be generally classified into two categories, Le., unstructured models and structured models. The former treats a culture as a lumped quantity of biomass and does not consider intracellular components; the latter considers the heterogeneity of a culture and the intracellular components (Bailey and Ollis, 1986). Despite impressive progress made recently in developing structured models for microbial growth (Nielsen and Villadsen, 1992), the unstructured models or semimechanistic models are still the most popular ones used in practice. The unstructured models include the most fundamental observations concerning microbial growth and are simple and easy to use, particularly for process control purposes. The most widely used unstructured model for the substrate uptake rate is the Monod equation:
* Corresponding 0049-531-6181 11 1.
author: Telephone, 0049-531-6181188; Fax,
In eq 1,p = AC,) is the specific growth rate of cells and is expressed as a function of substrate concentration C, [for Monod kinetics p = ,uu,,,C$(C, K,); other forms are needed at limitations other than those of the substrate]. in eq 1was originally taken to The yield coefficient Yx/, be constant. However, it was realized later that the specific growth rate p may influence Yx/, under certain conditions. To account for this effect, Pirt (1965) introduced his well-known maintenance model:
+
T"
In eq 2, and m, are the maximum growth yield and maintenance requirement for substrate, respectively. A similar equation was introduced in 1973 by Stouthamer and Bettenhausen to describe the specific energy (ATP) requirement for microbial growth:
In eq 3, c g and mATP are the maximum energetic growth yield and maintenance energy requirement, respectively.
8756-7938/95/3011-0071$09.00/0 0 1995 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1995, Vol. 11, No. 1
72
The concept of a maintenance requirement was proposed by Pirt (1966) to account for the non-growthassociated energy requirement of microorganisms: for instance, the energy necessary for the turnover of cell wall substances and other macromolecules and for maintaining transmembrane ion gradients. For a number of anaerobic cultures, and particularly for growth under substrate-limited conditions, m, and mATP have been found to be independent of p (Stouthamer, 1977). However, it has also been shown that both m, and mATP may vary with p in some cases, particularly in aerobic cultures and under energy-suflicient conditions (Tempest and Neijssel, 1984; Stouthamer and van Veseveld, 1985;Tsai and Lee, 1990). Neijssel and Tempest (1976) modified eqs 2 and 3 to account for the influence of p on the maintenance requirement:
Substrate
Cell mass
and enemy a m )
Growth factors
Maintenanceenergy
1
energy
1
Energy spilling (uncoupling)
Figure 1. Simplified diagram of material and energy inputs and outputs of microbial metabolism.
Experimental data for three different cultures from the literature are used to verify the proposed kinetic model. where m‘ is the maintenance requirement (for substrate and energy) at p = 0 and c is an indeterminate constant that could be positive or negative. Pirt (1982) proposed a modification similar to eq 4 and claimed that m, of eq 2 decreases linearly with increases in p. More recently, Tsai and Lee (1990) considered the “overutilization” of the energy and carbon source in energy-sufficient cultures. These authors attributed the overutilization of the energy substrate to a defective regulation of the energy substrate consumption and energy uncoupling (spilling). They proposed that the overall specific substrate consumption rate of an energysufficient culture is the sum of the “true” substrate requirement qs*,which is the specific substrate consumption rate under energy-limited conditions (e.g., eq 21, and the specific substrate consumption by energy spilling, q E :
By introducing a variable energy surplus to describe the extent of energy excess, the following equation was derived:
4, = (1 - R k ) R s @ , x a X-k mS> +
&@eax + m,> (6)
+
where Rk = kl/(kl k d accounts for the effects of regulation and energy uncoupling and R , = C$(K, C,) accounts for the effect of substrate concentration. kl and kz are positive constants related to the regulation of substrate consumption by energy surplus and the rate of energy spilling, respectively. Some of the constraints of this model have been pointed out by Tsai and Lee (1990) and will be discussed further later in this work (cf. Model Development). One major shortcoming of eq 6 is that it describes the influence of substrate excess on 4,only at a relatively low substrate excess (i.e., at C, < ca. 5K,). It is, however, known for a number of cultures that the influence of substrate excess is still quite significant at much higher residual concentrations (Le,, even at C,s-lOK,) (Hueting and Tempest, 1979; Egli and Quayle, 1986; Brooke et al., 1990; Fuhrmann and Valker, 1992; Zeng et al., 1993). In this work, a kinetic model having the basic form of eq 5 is proposed to describe the substrate and ATP consumption rates of microbial growth under sufficient substrate (carbon excess), in which the substrate and ATP consumption rates are not only a function of the specific growth rate but also a function of the residual substrate for a relatively broad concentration range.
+
Model Development Substrate-SUtticientCulture Growth. The growth of microorganisms requires the input of a carbon source (substrate), energy, growth factors, and a number of other nutrients. For some microorganisms (e.g., chemoautotrophs), the energy and carbon required for growth are from two different sources. In this study, we consider only microbial growth with a single substrate, which is used at the same time as the carbon and energy source. Figure 1shows a simplified diagram of the material and energy inputs and outputs involved in microbial growth. Depending on the relative availability of substrate and nutrients (including growth factors), the growth of cells may be limited by the energy substrate (including oxygen for strict aerobes) or by nutrients such as nitrogen or phosphorus. In some cases cell growth may be partly or completely limited by product inhibition. If growth is limited by the availability of the carbon source, it is said to be substrate-limited. If growth is limited by other nutrients or by product inhibition and the carbon source is, relatively speaking, in excess, this type of cell growth will be called substrate-sufficient. From the viewpoint of energetics, microbial growth may be also classified into energy-limited and energy-sufficient growth (Tsai and Lee, 1990). However, the latter classification is, in some cases, confusing since some seemingly nutrient-limited cultures may actually be energy-limited and vice versa (Neijsel and Tempest, 1976). Furthermore, no exact mathematical form can be given for the definition of energy-limited or energy-sufficient cultures. For substrate-limited cultures without product inhibition, the Monod equation may be used to describe the relationship between substrate concentration and the specific growth rate:
From eq 7, the critical substrate concentration €or substrate-limited growth at a given growth rate can be defined as
If the maximum growth rate pmsxand the saturation constant K,of a culture are known, eq 8 can be used as a criterion to determine whether a culture is substratelimited or substrate sufficient. If the residual substrate concentration of a culture is obviously higher than the
Biotechnol. Prog., 1995, Vol. 11, No. 1
73
0 Potassium limitation h Phosphate limitation
1 "
I
0,O
I '
1
0,l
.
I
0,2
.
I
0,3
'
I
0,4
'
I
0,5
'
I
0,6
Specific growth rate p ( h-' ) Figure 2. Specific glucose consumption rates of K. aerogenes in anaerobic chemostat cultures under different limitations (pH 6.8, 35 "C) [data from Teixeira de Mattos and Tempest (198311.
critical substrate concentration, as calculated from eq 8, the culture is said to be substrate-sufficient. It should be mentioned that the term "dual or multiple limitation" is sometimes used in literature to refer to limitations by the energy substrate and other nutrients simultaneously. Given that ,uma and K, in eq 8 are constant, it is clear that under these so-called dual or multiple limitation conditions the residual substrate concentration must be higher than that calculated by eq 8. According to the above definition, this type of culture also belongs to the substrate-sufficient cultures. This also applies to situations where product inhibition exists. Substrate and Energy Consumption under Different Limitations. As illustrated in Figure 1, the substrate taken up by a microorganism is metabolized to form various intracellular metabolites and energy (ATP). The metabolites and energy are used for biomass formation, maintenance, and product formation. The metabolites and energy may also be consumed by energy spilling or through futile cycles. The consumption rate of the energy and carbon source is determined by the rate a t which the extracellular substrate enters the cell and how fast the substrate is metabolized intracellularly. The uptake of a substrate can be achieved by different uptake mechanisms, i.e., passive diffusion, facilitated diffusion, group translocation, proton-linked transport, and binding-protein-dependent transport (Gottschalk, 1986). Often, more than one transport system may be involved simultaneously in transporting one substrate. The activities of these transport systems are under the feedback control of many intracellular molecules. In higher animals, substrate consumption may be strictly regulated to match the energy requirement. In microorganisms, however, the regulation of substrate uptake may not be effective, for example, in the case of glucose metabolism in growing yeast cells (Fiechter and Seghezzi, 1992). Tempest and Neijssel(l984) and Tsai and Lee (1990) discussed the substrate (glucose) utilization of microorganisms under different limiting conditions. In general, substrate-sufficient cultures have a much higher substrate consumption rate than substrate-limited cultures a t the same specific growth rate, particularly a t low growth rates. Figure 2 shows a typical comparison of the substrate consumption rates of Klebsiella aerogenes cultures grown anaerobically on glucose with different
limitations. Similar trends were observed with K. aerogenes grown aerobically on glucose (Neijssel and Tempest, 1976) and other microorganisms (Egli and Quayle, 1986; Brooke et al., 1990; Zeng et al., 1993). All of these results cannot be fitted conveniently with eqs 2 and 3 and their modified form eq 4, or when fitted the parameters r"" and m obtained are difficult to interpret. Tempest and Neijssel(1984) thus questioned the status of Y"" and m as biologically interpretable phenomena. Qualitatively, it was argued that the high substrate (energy) consumption rate of potassium-, phosphate-, and ammonia-limited cultures is mainly due to the high energy requirement for maintaining ion gradients across the cytoplasmic membrane (Neijssel and Tempest, 1976; Hueting et al., 1979). Other reasons may include futile cycles and energy spilling, metabolic uncoupling, and modification of the respiratory chain under energysufficient conditions. In addition, when the energy and carbon source is in excess, microorganisms tend to dispense the excess energy and carbon by the formation of storage compounds or extracellular products. Some of the products such as acetic acid and ethanol are toxic to cell growth. Growth ultimately may be inhibited by the accumulation of such metabolites. Inhibition by products can exert stress on the cell machinery, and more energy (substrate) is required to overcome it, as is shown for growth inhibition by acetic acid (Zeng et al., 1990a). The metabolites may also act as an energy uncoupler, thus increasing the substrate consumption rate. Furthermore, a large number of microorganisms can produce a variety of products having different energetic efficiencies and different toxicities. The selectivity of these products is offen altered under different limiting conditions. Under substrate-limited conditions, regulation of the enzyme systems may allow cells to form products with high energy efficiency. On the other hand, when substrate is sufficient, cells may prefer pathways leading to less toxic products in order to maximize cell growth (Zeng et al., 1993). This may also lead to a lower energetic efficiency of substrate conversion. Thus, more substrate is required to obtain the same amount of energy production for growth. Effect of Excessive Substrate on the Substrate and Energy Consumption Rate. From the above discussion, it is evident that substrate and energy consumption rate of microbial growth under substratesufficient conditions could be significantly higher than that under substrate-limited conditions. Figure 3 shows the effects of excessive glucose on the consumption of glucose by K. aerogenes grown aerobically in an ammonialimited culture a t a constant dilution rate (Hueting and Tempest, 1979). Also shown in Figure 3 is the specific ATP consumption rate, qATP. qATp is calculated with the following equation, which can be derived in a way similar to that used by Zeng et al. (1990b) for Enterobacter aerogenes: qATP
= 2(p/o)qOz
+ '/3qCOZ + 4/3qHAc + q p y r + 4/3qketoglu (9)
where gozrqcOn,qmC, qPF,and qketoglu are the consumption and/or formation rates of oxygen, carbon dioxide, acetate, pyruvate, and ketogluconate, respectively. P/O is the energy efficiency of oxidative phosphorylation taken as 1.75 mol of ATP/O.5 mol of 0 2 , as found for E. aerogenes (Zeng et al., 1990b). With an input ammonia concentration fixed at 30 m m o w and an inlet glucose concentration of less than 7.5 g/L, cell growth was initially glucose-limited and the specific rates of glucose and ATP consumption were
Biotechnol. Prog., 1995, Vol. 11, No. 1
74
Considering the resulta shown in Figure 3, it is obvious that qsEand qATpE should have the form of MichaelisMenten or Monod kinetics. In order to keep the models simple and practicable, the mechanisms for the increased substrate and energy consumption rates are not considered here. Rather, the following semiempirical models are proposed for qsEand qATPE:
0
10 20 Input glucose concentration @/I)
30
Figure 3. Influence of input glucose concentration on the consumption rates of substrate and ATP for K. aemgenes in an
ammonia-limitedaerobic chemostat culture at a constant dilution rate of 0.40 h-l [data from Hueting and Tempest (197911.
constant. Increasing the input glucose concentration above this value (cf. Figure 3) led to sharp increases in q, and qATp, and the excretion of partidly oxidized metabolites was observed (Hueting and Tempest, 1979). The culture was obviously substrate-suflicient when glucose appeared in the culture in noticeable concentrations. Under these conditions, q, and qATp increased markedly at low residual glucose concentrations but leveled off at high residual glucose concentrations, indicating a saturation of substrate uptake. Similar results were reported for Hansenula polymorpha under ammonia-limited conditions (Egli and Quayle, 19861, for Bacillus strain TS1 grown on methanol under nitrogenand potassium-limited conditions (Brooke et al., 1990), for K. pneumoniae grown on glycerol under productinhibited conditions (Tag, 1990;Zeng et al., 19931,and for the biodegradation of phenol by activated sludge (Beltram et al., 1984). Mathematical Models. In a manner similar to that of Tsai and Lee's model, the specific rates of consumption of substrate and energy (q,* and qATp*) under substratelimited conditions are considered the minimum substrate and energy requirements of a substrate-sufficient culture at the same specific growth rate. At steady state, the overall specific rates of consumption of substrate and energy are the sums of the minimum requirements and their excessive consumption rates (asEand qATpE) due to excess substrate:
with QS*
=
& + m, 1 ATP
In eqs 14 and 15, Aqra and AqEg are the maximum excess rates of substrate and energy consumption under substrate-sufficient conditions, respectively. They may depend on the nature of nutrient limitation, but they are assumed to be constant for a given type of limitation. C,* is the residual substrate concentration of the corresponding substrate-limited culture at the same specific growth rate and can be calculated from eq 8. Thus, C, - C,* = AC, represents the excess of substrate concentration.K,* and KAT^* are saturation constants. Since C,* is a function of p, qSEand qATpE may also be functions of p. Substitution of eqs 12-15 into eqs 10 and 11 gives
c, - c,*
C,- C,*+ KATp* C, L C,* (17) Equations 16 and 17 imply that the substrate and energy consumption of cells under substrate-sutlicient conditions consists of three parts, Le., the growth-independent maintenance requirement (m), the growth rate-dependent consumption gl/ymax), and the "spilling" due to substrate sufficiency [qE= Aqmax(Cs- C,*)/(C, - C,*
IWl.
+
Comparisonwith Existing Models. At low residual eqs 16 and 17 substrate concentration (C,- C,**:Ks*), reduce to Pirt's well-known linear equation for substrate consumption (eq 2)and Stouthamer and Bettenhausen's linear equation for ATP consumption (eq 31, respectively. At high residual substrate concentration (C,- C,* * Ks*), eqs 16 and 17 reduce to
As Aqrax and Aqzg are constants for a given type of culture, eqs 18 and 19 imply that at high excess substrate Concentration the rates of substrate and energy consumption are also a linear function of the culture growth rate. As shown in Figure 2 for K. aerogenes, the slopes of q, vs p at different limitations do not differ much, indicating that the same maximum substrate and ATP yields can be used. However, the non-growth-associated part of the substrate and energy consumption rates is increased by
Biotechnol. Prog., 1995, Vol. 11, No. 1
75
Aqr" and AqF& respectively, compared to the culture under substrate limitation. The terms m, Aqr" and
+
+
mATp AqFg represent the maximum rates of substrate and energy consumption due to maintenance and other non-growth-associated cellular functions and depend on the nature of the limitation. At moderate residual substrate concentration, the substrate and energy requirements for non-growth-associated functions increase with increasing excess substrate. These dependencies of the maintenance requirement may well explain the often observed variations in the maintenance requirement under different conditions and sometimes seemingly controversial reports by different authors on the maintenance requirement of the same microorganism under similar conditions (Tempest and Neijssel, 1984; Stouthamer and van Verseveld, 1985). It also illustrates the importance of medium design when dealing with the substrate and energy yields of microbial growth. At C, = 0 (K,* > K, >> Cs),p must also approach zero. From eqs 16 and 17 we have q, = m, K,* > K, >> C,
(20)
Thus, m, and mATP can be considered to be the substrate and energy consumption rates required for cells to avoid cell death or lysis, respectively (i.e., to sustain p 2 01, a t C, = 0. Compared to the Pirt model (including its modifications) and Tsai and Lee's model, the parameters in the proposed models are clearly defined, and the same maximum yield Ym" and maintenance term m can be used for different limitations and under different residual substrate concentrations, without the necessity of introducing the so-called apparent maximum yield and maintenance coefficient. This is in accordance with the experimental results obtained for K. aerogenes under different limitations (cf. Figure 2). If a growth model accounting for either product inhibition, e.g., the extended Monod equation (Levenspiel, 19801, or growth limitation by another nutrient (Bailey and Ollis, 1986) is used to calculate p, eq 16 can be written as q, = m,
Pmax +-
c, - C," Aqr" C, - C,*
+ K,"
(22a)
for the case of product inhibition, in which Ci is the concentration of the inhibitor and Ci* and n are constants, and
r.
(23) where ~ c , = ois the specific rate of biomass decrease after completion of substrate and may be measured directly. needs to be In order to calculate m, from eq 23, Tax estimated separately (see below). (2) and Aq,"". To estimate and Aqf", continuous cultures need to be run a t a series of dilution rates under excessive substrate concentration (C, - C," >> K,*). Again, a plot of q, against p should give a straight line under these conditions. The slope of this line is equal to l e " ; the intercept is equal to m, Aqr". With a known value of m,, Aq;" can then be calculated. It should be mentioned that the estimation of Aqr" may involve some trial and error procedures, because the condition C, - C,* >> K,* cannot a priori be ascertained before Ks*is determined. Thus, only those steady states should be used that give a straight line when plotting q, vs p. (3) Estimation of Ks*. To estimate K,*, experiments should be run at different dilution rates with variation of the substrate Concentration in the medium to ensure a varying residual substrate concentration in the culture. Equation 16 can be rearranged to the following form:
T"
+
+
C, ca r a x K s+ Cs Ka + Ca
Pmax
qs=ms+---
of substrate concentration over a relatively narrow range up to about C, I5K,. This is obviously too low for the range of significance reported in the literature (Hueting and Tempest, 1979; Egli and Quayle, 1986; Brooke et al., 1990; Zeng et al., 1993). Also, the parameters in the proposed model can be experimentally evaluated as shown in the following section. Evaluation of Model Parameters. Theoretically, the parameters in eqs 16 and 17 can be estimated by fitting experimental data. Nevertheless, problems may be encountered in some cases. In the following section, graphical methods are proposed to evaluate the model parameters separately. For the evaluation of the parameters in eqs 16 and 17, the same methods as those proposed here can be used. Therefore, only those for the substrate uptake model are presented. In order to estimate these two (1) m, and parameters, continuous cultures can be run under substrate limitation at a series of different dilution rates. According to eq 16, a plot of q, against should give a straight line under these conditions. The slope of this line is equal to l/ka"=; the intercept is equal to m, (Pirt, 1975). m9may also be measured directly under conditions of C, = 0 and q, = 0. Under these conditions, it follows from eq 16 that
(22b)
According to eq 24, plotting Aqf"/(q, - m, - p r ) vs 14C, - C,*) should give a straight line. The slope of this line is equal to K,*.
for the case of multiple nutrient limitation, in which C, is the concentration of the alternatively growth-limiting nutrient and K, is its Monod saturation constant. In both cases, the proposed model uses two saturation constants for the substrate and can thus describe the effect of substrate concentration over a wide range. Whereas Pirt's model does not consider the substrate concentration a t all, Tsai and Lee's model can only describe the effect
Results and Discussion Anaerobic Growth of Klebsiella pneumoniae on Glycerol. The anaerobic conversion of glycerol to 1,3propanediol is a bioprocess of industrial interest and recently has been receiving more and more attention. Continuous cultivation of K. pneumoniae on glycerol had been carried out by Tag (1990) at different dilution rates in three serial experiments with different initial glycerol
c, - c,*
Aqrax C, - C,*
+ K,"
Biolechnol. Prog., 1995, Vol. 11, No. 1
76
A
C,> 10 K*,
0 C, 100 mmol/L). It can be seen that the specific substrate consumption rate of K. pneumoniae is significantly higher under excessive substrate conditions than under substrate-limited conditions. A similar trend can be shown for the specific ATP consumption rate. In Figure 4, two straight lines with the same slope are obtained. According to the proposed methods for parameter evaluation, the parameters ma, and AqrQ for K. pneumoniue can be determined from the slopes and intercepts of these lines, having the values 1.2 mmoVgh, 0.0078 g/mmol, and 34.6 mmoVgh, respectively. Figure 5 shows a plot of Aqrax/(qs- m, p e a ) vs UC, - C,*). Only steady states with C, obviously higher than C,* were used in order to avoid possible errors in the determination of C, at very low concentrations. Although the data are found to scatter somewhat a straight line could be drawn that gives a K,* value of about 7.0-8.0 mmoVL. These results demonstrate that the proposed q, model and methods for parameter estimation can be adequately applied to the glycerol fermentation by K. pneumoniae. Equation 16, with the parameters estimated above, describes the q, values of 16 steady states with an average deviation of 5.28%. We noticed that a simultaneous estimation of all of the parameters by fitting the
c
cm,
Figure 5. Graphical estimation of the parameter K,* for K. pneumoniae grown anaerobically on glycerol.
experimental data with eq 16 gives slightly different parameters but nearly the same average deviation. The resulting equation for q8 thus determined is as follows:
c, - C," 32.9 c, - c,* 9.3 (mmol/gh) C, L C,* (25)
+
A simultaneous estimation of parameters in eq 17 resulted in the following qATp equation for K. pneumoniae:
c, - C," C, - C," + 12.3 (mmol/gh) C,L C,* (26) Equations 25 and 26 describe the glycerol fermentation by K. pneumoniae with average deviations of 5.21%and 7.12%,respectively, which reduce to 3.91%and 5.21%if steady states at the lowest dilution rate (0.08 h-l) are not considered. The K,*and KAT^* values obtained are close to each other, indicating that the responses of K. pneumoniae to substrate excess are essentially the same in terms of substrate and energy consumption. These values for K,*and KAT^* are 30-50-fold higher than the K,value (0.26 mmoVL). Hence, C,* in eqs 25 and 26 may be omitted without a noticeable influence on qsE and qATpE.
Figure 6 shows the excess substrate and ATP consumption rates as functions of the residual glycerol concentration at different dilution rates. Both qsEand qATpE are only functions of the residual substrate concentration, irrespective of the specific growth rate. The simulations as calculated from eqs 25 and 26 agree reasonabely well with the experimental results and confirm the applicability of the proposed kinetic models (eqs 16 and 17). Aerobic Growth ofB d U w , Strain Ts1 on Methanol. MethylotrophicBacillus strains are novel methanol utilizers that may be of considerable industrial interest both as producers of thermostable enzymes and for transformations to useful metabolites. Brooke et al.
Biofechnol. Prog., 1995, Vol. 11, No. 1
77
12
20
40
DS0.47
" Substrate
'=g]
D=O 40
D.0.27
30
15
D=0.38
A
....' 0 50 100 150 200 250 300 350 400 Residual substrate (mmol glycerolll)
' . I . . . . , . . . . I . . . . I . . . . I . . . . I . . . ' I
0
0,O 0,l
(1989, 1990) studied the metabolic fluxes of Bacillus strains in methanol-limited and methanol-sufficient chemostat cultures. Figure 7 shows the specific consumption rates of methanol and oxygen of the Bacillus strain TS1 as a function of dilution rate in a carbonlimited culture. Linear relationships are found for both qsand qo2under these conditions. hsuming q A T P = (2(P/ 0) 2/3)qo2(under conditions without product formation), QATp obviously is also a linear function of p under methanol limitation. Since the (P/O) ratio for this peculiar Bacillus strain is not known, the kinetics for qoz instead of for q A T P is considered here for substratesufficient cultures. Provided that the (P/O) ratio is constant and product formation does not contribute appreciably to energy generation, the consumption rate of oxygen should be an indirect variable for the consumption rate of energy. Thus, qATP should have a form similar to that of qo2. Brooke et al. (1990) studied the effect of increasing the input methanol concentration on the metabolism of Bacillus strain TS1 in nitrogen- and potassium-limited chemostat cultures at a constant dilution rate (D = 0.20 h-l). Figures 8 and 9 show the excess consumption rates of methanol and oxygen as functions of the residual methanol concentration in the culture. Obviously, the increments of q, and qo2 with increasing C, follow the proposed kinetic models (eqs 14 and 15) under both nitrogen and potassium limitations. The following equations are established by data fitting for the nitrogenlimited culture:
+
(27)
c, - c,*
c, - c,*+ 3.93
C, 1 C,* (28)
and for the potassium-limited culture: q: = 23.35
c, - c,* C, C, - C,* + 3.90
qoZE= 28.82
c, - c,*+ 1.22 C, 2 C,*
L
C,*
0,5
0,6 0,7 0,8
Dilution rate (h" )
Figure 6. Effect of residual glycerol concentration on the excess of glycerol and ATP consumption rates of K.pneumoniae at h-1) in continuous culture. different dilution rates (D,
qoaE= 11.86
0,2 0,3 0,4
(29)
(30)
Figure 7. Specific consumption rates of methanol and oxygen of Bacillus strain TS1 as a function of dilution rate in carbonlimited chemostat culture [pH 6.8, 52.5 "C; data from Brooke et al. (198911.
As shown in Figures 8 and 9, eqs 27-30 describe the experimental data satisfactorily, with average deviations of 5.5% and 4.4% for q, under nitrogen and potassium limitations, respectively, and 6.7% and 3.6% for qo2. In the preceding calculations, the C,* value at D = 0.20 h-l was estimated by data fitting. The same values for C,* (ca. 1.16 mmoVL) were obtained from both the q, and qo2 data sets. Aerobic Growth of Klebsiella aemgenes on Glucose. Tempest and his coworkers (Neijssel and Tempest, 1975, 1976; Hueting and Tempest, 1979; Hueting et al., 1979; O'Brien et al., 1980) systematically studied the aerobic growth of K. aerogenes on glucose under welldefined conditions with different limitations. Under glucose-limited conditions, the consumption rates of glucose, oxygen, and hence ATP were found to be linear functions of the specific growth rate. Under glucosesufficient conditions, enhanced consumption of glucose and oxygen was observed. Typical results of the influence of glucose input concentration on the substrate and energy consumption of K. aerogenes are shown in Figure 3 for an ammonia-limited chemostat culture at a constant dilution rate (D = 0.40 h-l) (Hueting and Tempest, 1979). Figure 10 shows the increment of q, as a function of the residual substrate concentration in the culture. Also shown in Figure 10 are the results of data fitting with the proposed model (eqs 14), which gives the following equation: q: = 3.24
c, - c,* C, - C,*
+ 0.644 C, L C,*
(31)
The model describes the experimental data reasonablly well, with average errors of 4.4%. The K,* value was found to be quite low, indicating that the glucose effect is significant at relatively low substrate excess. However, compared to the K, value (0.01 mmoyL) reported for K. aerogenes under the experimental conditions (O'Brien et al., 19801, the estimated K,*value is 64-fold higher than K,. The excess qATP seems to be more sensistive to the substrate excess (cf. Figure 3). Unfortunately, no data were available for the residual glucose at input glucose concentrations below 10 g/L. Thus, no reliable correlation could be obtained for qATpE under these conditions.
70
Biotechnol. Prog., 1995, Vol. 11, No. 1
-181
0 Experimental
0 ExperlmMltel
-Calculated
a
Calculated 16
-? P
"
"
,
.
'
,
'
I
.
20
0
,
.
.
I
40
,
,
,
.
I
.
,
,
.
0
.
80
60
14-
20
0
100
40
80
60
1
Residual substrate (mmol methanolll)
Residual substrate (mmol methanolll) 60
0 Experimental
b
55
-Calculated
9 40-
g
35:
V
g +-
25-
5
20:
-t
15,
E
18
.
30-
lo]/ 5
o ,
0
~
5
r
,
10
.
15
[
r
~
20
25
,
l
30
.
~
35
,
l
40
.
o f # J . l ~ l . l . l , , . , . l , 0
5
10
15
20
25
30
35
Residual substrate (mmoVI) Figure 8. (a) Effect of residual methanol concentrationon the increment of methanol consumption rate of Bacillus strain TSl in nitrogen-limited chemostat culture (D= 0.20 h-l) [data from Brooke et al. (199011. (b) Effect of residual methanol concentration on the increment of methanol consumption rate of Bacillus strain TS1 in potassium-limited chemostat culture (D= 0.20 h-l) [data from Brooke et al. (199011.
Residual substrate (mmol/l) Figure 9. (a) Effect of residual methanol concentrationon the increment of oxygen consumption rate of Bacillus strain TS1 in nitrogen-limited chemostat culture (D= 0.20 h-l) [data from Brooke et al. (1990)l. (b) Effect of residual methanol concentration on the increment of oxygen consumption rate of Bacillus strain TS1 in potassium-limited chemostat culture (D= 0.20 h-l) [data from Brooke et al. (1990)l.
Considering the experimental results shown in Figure 3, it is evident that qATpE should also follow the proposed kinetic model, probably with &p* lower than K,*.As shown in the work of Hueting and Tempest (1979),the potassium-limited culture seems to be even more sensitive to excess glucose.
responding specific growth rate and an gdditional consumption rate due to excess substrate. The model also applies to the specific ATP consumption rate and to the specific oxygen consumption rate of an aerobic culture. The linear equations for substrate and ATP consumption rates of Pirt and of Stouthamer and Bettenhausen can be used for substrate-limited growth. The excess substrate and ATP consumption rates were found to be h c t i o n s of the residual substrate concentration and can be expressed in a form similar to that of MichaelisMenten or Monod kinetics. The proposed model satisfactorily describes the effects of excessive substrate concentrations on the substrate and ATP consumption rates of three different cultures under a variety of growth limitations. Using this model, the maximum growth yield and maintenance requirement (both in terms of substrate and ATP) of a culture
Conclusion
An extended kinetic model for the specific rates of substrate and energy consumption has been proposed for the growth of heterotrophic microorganisms under substrate (carbon and energy source)-sufficientconditions. In ita convenient form, the specisc substrate consumption rate of a substrate-sufficient culture can be expressed as the sum of the substrate consumption rate of microorganisms under substrate-limited conditions at the cor-
Biotechnol. Prog,, 1995, Vol. 11, No. 1
Residual Substrate (mmol glumsell)
Figure 10. Effect of residual glucose concentration on the increment of substrate consumption rate of K.aerogenes in an ammonia-limited chemostat at a constant dilution rate of 0.40 h-l [data from Hueting and Tempest (197911.
under different growth limitations can be better defined and quantified. The range of residual substrate concentrations in which the specific rates of substrate and ATP consumption are affected can also be assessed. This information should be helpful in designing medium and reactor operating conditions and in interpreting experimental results obtained under different conditions.
Literature Cited Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2nd ed.; McGraw-Hill: New York, 1986. Beltram, P.; Beltram, P. L.; Carniti, P. Influence of feed concentration on the kinetics of biodegradation of phenol in a continuous stirred reactor. Water Res. 1984, 18 (41, 403407. Brooke, A. G.; Watling, E. M.; Attwood, M. M.; Tempest, D. W. Environmental control of metabolic fluxes in thermotolerant methylotrophic Bacillus strains. Arch. Microbiol. 1989, 151, 268-273. Brooke, A. G.; Attwood, M. M.; Tempest, D. W. Metabolic fluxes during the growth of thermotolerant methylotrophic Bacillus strains in methanol-sufficient chemostat cultures. Arch. Microbiol. 1990, 153, 591-595. Egli, Th.; Quayle, J. Influence of the carbon:nitrogen ratio of the growth medium on the cellular composition and the ability of the methylotrophic yeast Hansenula polymorpha to utilize mixed carbon sources. J. Gen. Microbiol. 1986, 132, 17791788. Fiechter, A.; Seghezzi, W. Regulation of glucose metabolism in growing yeast cells. J. Biotechnol. 1992,27, 27-45. Fuhrmann, G. F.; Volker, B. Regulation of glucose transport in Saccharomyces cerevisiae. J . Biotechnol. 1992,27, 1-15. Gottschalk, G. Bacterial Metabolism, 2nd ed.; Springer: New York, 1986. Hueting, S.; Tempest, D. W. Influence of the glucose input concentration on growth kinetics of metabolite production by Klebsiella aerogenes NCTC 418: growing in chemostat culture in potassium- or ammonia-limited environments. Arch. Microbiol. 1979, 123, 189-194.
79 Hueting, S.; de Lange, T.; Tempest, D. W. Energy requirement for maintenance of the transmembrane potassium gradient in Klebsiella aerogenes NCTC 418: a continuous culture study. Arch. Microbiol. 1979, 123, 183-188. Levenspiel, 0. The Monod equation: A revisit and a generalization product inhibition situations. Biotechnol. Bweng. 1980, 22, 1671-1687. Neijssel, 0. M.; Tempest, D. W. The regulation of carbohydrate metabolism in Klebsiella aerogenes NCTC418 organisms growing in chemostat culture. Arch. Microbiol. 1975, 106, 251-258. Neijssel, 0. M.; Tempest, D. W. Bioenergetic aspects of aerobic growth of Klebsiella aerogenes NCTC418 in carbon-limited and carbon-sufficient chemostat culture. Arch. Microbiol. 1976,107, 215-221. Nielsen, J.;Villadsen, J. Modelling of microbial kinetics. Chem. Eng. Sci. 1992,47,4225-4270. Neijssel, 0. M.; Hueting, S.; Crabbendam, K. J.; Tempest, D. W. Dual pathways of glycerol assimilation in Klebsiella aerogenes NCTC 418: Their regulation and possible functional significance. Arch. Microbwl. 1975, 104, 83-87. O'Brien, R. W.; Neijssel, 0. M.; Tempest, D. W. Glucose phosphoenolpyruvatephosphotransferase activity and glucose uptake rate of Klebsiella aerogenes growing in chemostat culture. J. Gen. Microbiol. 1980, 116, 305-314. Pirt, S. J. The maintenance energy of bacteria in growing cultures. Proc. R . SOC. London, Ser. B 1965, 163, 224-231. Pirt, S. J. Principles of microbe and cell cultivation; Blackwell Scientific Publications: Oxford, U.K., 1975. Pirt, S. J. Maintenance energy: a general model for energylimited and energy-sufficient growth. Arch. Microbio2. 1982, 133, 300-302. Stouthamer, A. H. Energetic aspects of the growth of microorganisms. Symp. SOC.a n . Microbiol. 1977, 28, 285-315. Stouthamer, A. H.; Bettenhausen, C. W. Utilization of energy for growth and maintenance in continuous and batch cultures of microorganisms. A reevaluation of the method for the determination of ATP production by measuring molar growth yields. Biochim. Biophys. Acta 1973,301, 53-70. Stouthamer, A. H.; van Verseveld, H. W. Stoichiometry of microbial growth. In Comprehensive Biotechnology; MooYoung, M . , Ed.; Pergamon: New York, 1985; pp 215-218. Tag, C . G. Mikrobielle Herstellung von 1,3-Propandiol. Ph.D. Thesis, University of Oldenburg, Oldenburg, Germany, 1990. Tempest, D. W.; Neijssel, 0. M. The status of YAW and maintenance energy as biologically interpretable phenomena. Annu. Rev. Microbiol. 1984, 38, 459-486. Tsai, S. P.; Lee, Y. H. A model for energy-sufficient culture growth. Biotechnol. Bioeng. 1990, 35, 138-145. Zeng, A. P.; Biebl, H.; Deckwer, W.-D. Effect of pH and acetic acid on growth and 2,3-butanediol production of Enterobacter aerogenes in continuous culture. Appl. Microbiol. Biotechnol. 1990a,33,485-489. Zeng, A.-P.; Ross, A.; Deckwer, W.-D. A method to estimate the efficiency of oxidative phosphorylation and biomass yield from ATP of a facultative anaerobe in continuous culture. Biotechnol. Bioeng. 1990b, 36, 965-969. Zeng, A.-P.; Biebl, H.; Schlieker, H.; Deckwer, W.-D. Pathway analysis of glycerol fermentation by Klebsiella pneumoniae: Regulation of reducing equivalent balance and product formation. Enzyme Microbwl. Technol. 1993, 15, 770-779. Zeng, A.-P.; Ross, A.; Biebl, H.; Tag, C.; Giinzel, B.; Deckwer, W.-D. Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnol. Bioeng. 1994, in press. Accepted July 5, 1994.@ @
Abstract published in Advance ACS Abstracts, September 1,
1994.
